A lot will depend on just how prepared you want to be.

The most common issue you'll likely face is a puncture, and to fix a puncture you need tyre levers, a patch kit or spare tube, a pump, and perhaps a bit of patience. Depending on just what setup you have you may be able to get away without even using levers (I fixed a flat with Specialized Borough CX tyres with no tools at all, but really struggled to work with Schwalbe Marathon Plus tyres).

A basic multi tool containing a bunch of allen keys (not sure what you US folks call them - hex keys or allen wrenches?) will probably cater for most of what you'll need on the trail. I've never needed anything more complex than a pump and a couple of allen keys. Personally I like to carry a couple of loose allen keys as well on the basis an individual key is easier to work with than a multitool, simply because it's less cumbersome. A cross-head screwdriver is also worth having although I've never actually needed one in the field.

If you break a spoke you might be thankful for a spoke wrench to adjust the tension of your wheel to stop it rubbing. If you've got disc brakes you probably won't need to worry about that though.

You might want to take a few links of chain and a chain tool and maybe a couple of master links so you can deal with a chain breaking. I've never had that happen to me but I know people who have.

If you go much beyond that you have to decide where to stop. You could take some spare brake cables and shifter cables in case they break but unless you're into serious long-distance riding you might be better off learning how to improvise so you can get back to a decent workshop and fix it there.